Lewis Hardcastle's first half free-kick proved to be the difference between the two sides as Rovers made it a third semi-final in six seasons.
But the home side were pushed all the way by a spirited Luton side who were kept at bay by three excellent saves by Andy Fisher in the first half including a fingertip stop to touch a Jack Snelus effort onto the bar.
Rovers should have wrapped the game up in the second half but Stefan Mols, Lewis Mansell and Josh Askew all spurned great chances, which lead to a nervy ending to the match.
However the boys in blue-and-white held on to clinch a two-legged semi-final against either Chelsea or Reading.
Billy Barr made two changes from the side that knocked out Aston Villa in the previous round as Ramirez Howarth and Stefan Mols returned. Whilst the Hatters made just one from the team that defeated Preston North End on penalties as Kavan Cotter came in for Harry Bean.
It was the home side were strong starters at Ewood, taking the game to Luton Town from the off. Within 90 seconds Rovers had a sight of goal as Ryan Nyambe got around the back to feed Tre Pemberton but the winger lashed his left-footed shot over the bar.
Matty Platt then headed over a short corner from Lewis Hardcastle as the pressure was maintained, before defender Platt would be upended on the edge of the box on 12 minutes.
Hardcastle took the resulting free-kick, striking it low, the ball taking a deflection on it's way to the back of the net for his second goal in this season's FA Youth Cup.
Three minutes later and it was almost 2-0. A superb run from left-back Josh Askew saw the defender cut in from the byline, his cross striking a defender and bouncing off the foot of the post and behind.
After what had been a very one-sided opening, to their credit the visitors really came into the ascendency as the first half wore on. Andy Fisher making key saves to deny Jack Snelus and George Murray as Luton built up a head of steam.
Then after a Ramirez Howarth shot had been saved at one end, Luton were back on the attack at the other, Snelus letting fly from 25-yards, Fisher with vital fingertips to push the ball onto the crossbar.
Cameron McJannet then flashed a shot just wide of the far post as the Hatters looked to get back on level terms before the break, but it was Rovers who held the slender advantage as the half-time whistle blew.
After a scrappy opening to the second half it was Rovers with the opportunities to double their advantage.
Stefan Mols saw a shot deflect wide at the near post, from the resulting corner kick Mansell could only head straight at Liam Gooch in the Luton goal.
That prompted a double change from the Hatters as Harry Bean and James Verney were introduced, but it was Rovers sub Joe Grayson who fashioned the next chance.
His inswinging corner from the right was missed by Scott Wharton at the near post but again Mansell headed straight at the keeper.
As the game entered the closing stages play became increasingly stretched, with Rovers spurning further chances to put the game beyond the visitors.
First Mansell could only divert Askew's ball across the area wide of goal, before Grayson set the defender in on goal but his left-foot strike flashed past the far post.
With the game finely balance there was always the worry that Luton could grab a late equaliser and they had a trio of opportunties as the clock ticked down.
On 87 a high corner picked out Akin Famewo at the back post but the Luton captain headed wide. Then as the Hatters easily claimed possession in midfield Cotter fired straight into the arms of Fisher from distance.
There had been late drama in Rovers' last two Youth Cup matches this season, and as Kyran McGhan-George bore down on goal in the final moments it looked like another late goal was on the cards, however the young defender lost his footing at the vital moment and the chance was gone which meant it would be Rovers first through to the last four of the competition.
ROVERS: Fisher, Nyambe, Platt, Wharton (c), Askew, Hardcastle, Mols (sub Grayson 71 mins), Rankin-Costello, Pemberton (sub Fawns 84 mins), Mansell, Howarth.
Subs not used: Travis, Ascroft (GK), Steer.
Goals: Hardcastle 12 mins.
LUTON TOWN: Gooch, McGhan-George, Famewo (c), McJannet, Murray (sub Bean 69 mins), Cotter, Hinds (sub Verney 69 mins), Snelus, Atkinson, Justin, Bakinson.
Subs not used: Davey, Jones, Brown.